Question

The following problems consider the historic average cost per gigabyte of RAM on a computer. $$\begin{array}{|l|l|} \hline \text { Year } & \text { 5-Year Change (\$) } \\ \hline 1980 & 0 \\ \hline 1985 & -5,468,750 \\ \hline 1990 & -755,495 \\ \hline 1995 & -73,005 \\ \hline 2000 & -29,768 \\ \hline 2005 & -918 \\ \hline 2010 & -177 \\ \hline \end{array}$$ The average cost per gigabyte of RAM can be approximated by the function \(C(t)=8,500,000(0.65)^{t}\), where \(t\) is measured in years since 1980, and \(C\) is cost in USS. Find the average cost per gigabyte of RAM for 1980 to 2010 .

Short answer

The average cost per gigabyte of RAM decreased significantly from $8,500,000 in 1980 to approximately $22.67 in 2010.

Step-by-step solution

Understand the Function

The function given is \( C(t) = 8,500,000(0.65)^t \), where \( t \) represents the number of years since 1980, and \( C(t) \) is the cost per gigabyte of RAM in USD. This function models the cost decrease over time.

Calculate Cost for 1980

For 1980, \( t = 0 \) since 1980 is the base year. Substitute \( t = 0 \) into the function: \[ C(0) = 8,500,000(0.65)^0 = 8,500,000 \]. So, the cost in 1980 was $8,500,000 per GB.

Calculate Cost for 1985

For 1985, \( t = 5 \). Substitute \( t = 5 \) into the function: \[ C(5) = 8,500,000(0.65)^5 \]. Calculate the value: \[ C(5) = 8,500,000 imes 0.1160290625 = 986,246.53 \]. So, the cost in 1985 was approximately $986,246.53 per GB.

Calculate Cost for 1990

For 1990, \( t = 10 \). Substitute \( t = 10 \) into the function: \[ C(10) = 8,500,000(0.65)^{10} \]. Calculate the value: \[ C(10) = 8,500,000 imes 0.01365625 = 115,078.125 \]. So, the cost in 1990 was approximately $115,078.13 per GB.

Calculate Cost for 1995

For 1995, \( t = 15 \). Substitute \( t = 15 \) into the function: \[ C(15) = 8,500,000(0.65)^{15} \]. Calculate the value: \[ C(15) = 8,500,000 imes 0.00161501 = 13,727.585 \]. So, the cost in 1995 was approximately $13,727.59 per GB.

Calculate Cost for 2000

For 2000, \( t = 20 \). Substitute \( t = 20 \) into the function: \[ C(20) = 8,500,000(0.65)^{20} \]. Calculate the value: \[ C(20) = 8,500,000 imes 0.000191076 = 1,624.156 \]. So, the cost in 2000 was approximately $1,624.16 per GB.

Calculate Cost for 2005

For 2005, \( t = 25 \). Substitute \( t = 25 \) into the function: \[ C(25) = 8,500,000(0.65)^{25} \]. Calculate the value: \[ C(25) = 8,500,000 imes 0.00002259075 = 190.021 \]. So, the cost in 2005 was approximately $190.02 per GB.

Calculate Cost for 2010

For 2010, \( t = 30 \). Substitute \( t = 30 \) into the function: \[ C(30) = 8,500,000(0.65)^{30} \]. Calculate the value: \[ C(30) = 8,500,000 imes 0.0000026659061 = 22.667 \]. So, the cost in 2010 was approximately $22.67 per GB.

Key concepts

Average Cost Calculation

Understanding how to calculate the average cost per gigabyte of RAM over specific years involves breaking down and interpreting a mathematical function. The process begins by using the given formula, which in this case is: \[ C(t) = 8,500,000 \times (0.65)^t \]. Here, \( C(t) \) represents the cost per gigabyte for a year denoted by \( t \), starting from 1980. This formula describes an exponential decay, where the cost drops significantly over time.
To find the average cost over time, we need to evaluate this function at different values of \( t \) corresponding to each year of interest: 1980, 1985, 1990, and so on up to 2010. Each calculation provides insight into how the price of RAM has decreased over these years.
Calculate a few values using the formula to grasp the average trend:

• For 1980: \( C(0) = 8,500,000 \) USD
• For 1985: \( C(5) \approx 986,246.53 \) USD
• For 2000: \( C(20) \approx 1,624.16 \) USD
• For 2010: \( C(30) \approx 22.67 \) USD

These values illustrate the drastic reduction in RAM costs over the decades.

Historical Data Analysis

Analyzing historical data, like the cost changes of RAM, is a critical skill for understanding trends and making predictions. In this exercise, the historical average cost data is presented for each 5-year interval starting in 1980.
This type of data allows us to see not just the progression of prices over time, but also the pace at which these changes occur. Initially, we observe a steep decline in cost between 1980 and 1985. This is indicative of rapid technological advancements and market shifts during that period.
Each 5-year change represents the decline in average cost from one interval to the next, showcasing the exponential nature of the reduction. By examining these patterns:

• The most significant drop occurred between 1980 and 1985, reflecting a nascent era in computer technology.
• The declines become less dramatic from 2000 onwards, suggesting stabilization as technology matures.

Understanding the historical data helps in predicting future trends and identifying key periods of technological breakthroughs.

Mathematical Modeling

Creating mathematical models, like the exponential decay model used here, is essential for making sense of real-world phenomena. These models transform a series of past observations into a predictive tool, allowing us to estimate values for different time points.
The model provided, \( C(t) = 8,500,000 \times (0.65)^t \), captures the declining cost of RAM over time, characterized by an exponential decay. This decay is defined in the model by:

• Base Year: 1980, where the cost is \( 8,500,000 \) USD.
• Exponential Factor: 0.65, indicating how quickly costs decrease each year.

Mathematical modeling helps make future projections possible. For instance, by applying this formula, we can estimate future costs beyond the given data range or validate the model against historical data. This enables stakeholders to make informed decisions in pricing, market strategy, or technology development.
Understanding and constructing mathematical models give students and professionals the tools needed to tackle complex issues with confidence and clarity.